Singular Limits of the Klein–Gordon Equation

Abstract

We establish the singular limits, including semiclassical, nonrelativistic and nonrelativistic-semiclassical limits, of the Cauchy problem for the modulated defocusing nonlinear Klein–Gordon equation. For the semiclassical limit, \({\hbar\to 0}\), we show that the limit wave function of the modulated defocusing cubic nonlinear Klein–Gordon equation solves the relativistic wave map and the associated phase function satisfies a linear relativistic wave equation. The nonrelativistic limit, c → ∞, of the modulated defocusing nonlinear Klein–Gordon equation is the defocusing nonlinear Schrodinger equation. The nonrelativistic-semiclassical limit, \({\hbar\to 0, c=\hbar^{-\alpha}\to \infty}\) for some α > 0, of the modulated defocusing cubic nonlinear Klein–Gordon equation is the classical wave map for the limit wave function and a typical linear wave equation for the associated phase function.